A087787 a(n) = Sum_{k=0..n} (-1)^(n-k)*A000041(k).

1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409

Offset: 0

Author

Vladeta Jovovic, Oct 07 2003

Keywords

• nonn

Comments

Essentially first differences of A024786 (see the formula). Also, a(n) is the number of 2's in the last section of the set of partitions of n+2 (see A135010). - Omar E. Pol, Sep 10 2008

From Gus Wiseman, May 20 2024: (Start)

Also the number of integer partitions of n containing an even number of ones, ranked by A003159. The a(0) = 1 through a(8) = 15 partitions are:

() . (2) (3) (4) (5) (6) (7) (8)

(11) (22) (32) (33) (43) (44)

(211) (311) (42) (52) (53)

(1111) (222) (322) (62)

(411) (511) (332)

(2211) (3211) (422)

(21111) (31111) (611)

(111111) (2222)

(3311)

(4211)

(22211)

(41111)

(221111)

(2111111)

(11111111)

Also the number of integer partitions of n + 1 containing an odd number of ones, ranked by A036554.

(End)

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000041, A024786, A135010, A138121, A141285.

The unsigned version is A000070, strict A036469.

For powers of 2 instead number of partitions we have A001045.

The strict or odd version is A025147 or A096765.

The ordered version (compositions instead of partitions) is A078008.

For powers of 2 instead of -1 we have A259401, cf. A259400.

A002865 counts partitions with no ones, column k=0 of A116598.

A072233 counts partitions by sum and length.

Cf. A000009, A003159, A026804, A027187, A027193, A036554, A058695, A101707.

Programs

Mathematica

Table[Sum[(-1)^(n-k)*PartitionsP[k], {k,0,n}], {n,0,50}] (* Vaclav Kotesovec, Aug 16 2015 *)
(* more efficient program *) sig = 1; su = 1; Flatten[{1, Table[sig = -sig; su = su + sig*PartitionsP[n]; Abs[su], {n, 1, 50}]}] (* Vaclav Kotesovec, Nov 06 2016 *)
Table[Length[Select[IntegerPartitions[n], EvenQ[Count[#,1]]&]],{n,0,30}] (* Gus Wiseman, May 20 2024 *)

Python

from sympy import npartitions
def A087787(n): return sum(-npartitions(k) if n-k&1 else npartitions(k) for k in range(n+1)) # Chai Wah Wu, Oct 25 2023

Formula

G.f.: 1/(1+x)*1/Product_{k>0} (1-x^k).

a(n) = 1/n*Sum_{k=1..n} (sigma(k)+(-1)^k)*a(n-k).

a(n) = A024786(n+2)-A024786(n+1). - Omar E. Pol, Sep 10 2008

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 + (11*Pi/(24*sqrt(6)) - sqrt(3/2)/Pi)/sqrt(n) - (11/16 + (23*Pi^2)/6912)/n). - Vaclav Kotesovec, Nov 05 2016

a(n) = A000041(n) - a(n-1). - Jon Maiga, Aug 29 2019

Alternating partial sums of A000041. - Gus Wiseman, May 20 2024